If a particle moves from one point to another, we might need to know how fast it moves. Just as in Chapter 2, we can define two quantities that deal with “how fast”: average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation.

If a particle moves through a displacement in a time interval , then its average velocity is or

	(4-8)

This tells us that the direction of (the vector on the left side of Eq. 4-8) must be the same as that of the displacement (the vector on the right side). Using Eq. 4-4, we can write Eq. 4-8 in vector components as

	(4-9)

For example, if the particle in Sample Problem 4-1 moves from its initial position to its later position in 2.0 s, then its average velocity during that move is That is, the average velocity (a vector quantity) has a component of 6.0 m/s along the x axis and a component of 1.5 m/s along the z axis.

When we speak of the velocity of a particle, we usually mean the particle’s instantaneous velocity at some instant. This is the value that approaches in the limit as we shrink the time interval to 0 about that instant. Using the language of calculus, we may write as the derivative

	(4-10)

Figure 4-4 shows the path of a particle that is restricted to the xy plane. As the particle travels to the right along the curve, its position vector sweeps to the right. During time interval , the position vector changes from to and the particle’s displacement is .

FIGURE 4-4    The displacement of a particle during a time interval , from position 1 with position vector at time t1 to position 2 with position vector at time t2. The tangent to the particle’s path at position 1 is shown.

To find the instantaneous velocity of the particle at, say, instant t₁ (when the particle is at position 1), we shrink interval to 0 about t₁. Three things happen as we do so. (1) Position vector in Fig. 4-4 moves toward so that shrinks toward zero. (2) The direction of (and thus of ) approaches the direction of the line tangent to the particle’s path at position 1. (3) The average velocity approaches the instantaneous velocity at t₁.

In the limit as , we have and, most important here, takes on the direction of the tangent line. Thus, has that direction as well:

The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

The result is the same in three dimensions: is always tangent to the particle’s path.

To write Eq. 4-10 in unit-vector form, we substitute for from Eq. 4-1: This equation can be simplified somewhat by writing it as

	(4-11)

where the scalar components of are

	(4-12)

For example, is the scalar component of along the x axis. Thus, we can find the scalar components of by differentiating the scalar components of .

Figure 4-5 shows a velocity vector and its scalar x and y components. Note that is tangent to the particle’s path at the particle’s position. Caution: When a position vector is drawn, as in Figs. 4-1 through 4-4, it is an arrow that extends from one point (a “here”) to another point (a “there”). However, when a velocity vector is drawn, as in Fig. 4-5, it does not extend from one point to another. Rather, it shows the instantaneous direction of travel of a particle at the tail, and its length (representing the velocity magnitude) can be drawn to any scale.

FIGURE 4-5    The velocity of a particle, along with the scalar components of .